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Transcript of Basic Trigonometry and an Application

Trigonometry 3 functions we often use in physics Sine (abbr. sin) Cosine (abbr. cos) Tangent (abbr. tan) Yeah, I've heard of them but how do I use them? Let's visit our good friend the unit circle to learn more! Hey, that looks familiar but what's with the funny looking triangle? Better zoom in for a closer look! Okay, so I see x, y and theta but didn't this all start because you mentioned sine, cosine, and tangent? What's the connection? You'll see! OUCH! What could a boy getting his toe bit by a crab have to do with trig? How is he going to prevent infection? What? You heard me. How is he going to prevent infection? I have no idea. How? He's going to soh cah toa. Oh boy. He's got to "soak a toe-ah." Do you actually think that's clever? Yes, actually I do. :-) So, please tell me about SOH CAH TOA. SOH Sin = Opposite/Hypotenuse CAH Cos = Adjacent/Hypotenuse TOA Tan = Opposite/Adjacent I know what hypotenuse means! The hypotenuse is the longest side of a right triangle. It's called a right triangle because there is a right angle. The right angle is ALWAYS across from the hypotenuse and it measures 90 degrees. Oh wow! That red box represents the right angle. I hadn't noticed that before. So what are opposite and adjacent then? The opposite side is the side of the triangle which is across from the angle we want to talk about. So are you telling me, that if I was standing in the middle of the angle and walked right through the middle of the triangle, I'll run into the opposite side? Exactly! Cool! And doesn't adjacent mean next to (or something like that)? It sure does! So the adjacent side actually makes up part of the angle then along with the hypotenuse, right? You got it! Great! Let's try to apply this stuff then! Okay, let's look back at the triangle from our unit circle. For the marked angle,sin theta = y/1, cos theta = x/1and tan theta = y/x We can think of tangent as the rise over the run for the angle we are looking at.It's the opposite side (the rise) divided by the adjacent side (the run). What if the hypotenuse isn't conveniently 1 unit long? Then what? We will just use the length of our hypotenuse instead of using 1. We can come back to SOH CAH TOA to help us remember the formulas. I sure hope that kid's toe doesn't get infected! ;-P Hey! Wait a minute! The sine and cosine just have 1's on the bottom. Does that mean that for a unit circle, x = cos theta and y = sin theta? And the tangent formula kinda reminds me of the slope formula! Great connection! SOH: sin theta = opposite/hypotenuse CAH: cos theta = adjacent/hypotenuse TOA: tan theta = opposite/adjacent How is this going to help me with physics? If we are given the total velocity of a projectile (for example) and know its launch angle (theta), we can figure out it's x and y components using trig. sin theta = y velocity/total velocitywe can use to solve for y velocity:y velocity = total velocity * sin theta cos theta = x velocity/total velocity we can use to solve for x velocity:x velocity = total velocity * cos theta Ok. I think it makes sense! If I have any questions though I will be sure to ask! Projectile Motion Analyzing our projectile's motionbegins with.....